# On Small Sets of Integers

## TL;DR

This paper introduces a framework for small sets of integers based on upper quasi-densities, showing many interesting sets are small by linking them to the zero set of the upper Buck density.

## Contribution

It defines upper quasi-densities and upper densities on integers, characterizes small sets via the upper Buck density, and applies this to various number-theoretic sets.

## Key findings

- Small sets are characterized by their zero upper Buck density.
- Many number-theoretic sets are small, including those with limited prime factors.
- The framework unifies various density concepts for analyzing small sets.

## Abstract

An upper quasi-density on $\bf H$ (the integers or the non-negative integers) is a real-valued subadditive function $\mu^\ast$ defined on the whole power set of $\mathbf H$ such that $\mu^\ast(X) \le \mu^\ast({\bf H}) = 1$ and $\mu^\ast(k \cdot X + h) = \frac{1}{k}\, \mu^\ast(X)$ for all $X \subseteq \bf H$, $k \in {\bf N}^+$, and $h \in \bf N$, where $k \cdot X := \{kx: x \in X\}$; and an upper density on $\bf H$ is an upper quasi-density on $\bf H$ that is non-decreasing with respect to inclusion. We say that a set $X \subseteq \bf H$ is small if $\mu^\ast(X) = 0$ for every upper quasi-density $\mu^\ast$ on $\bf H$. Main examples of upper densities are given by the upper analytic, upper Banach, upper Buck, and upper P\'olya densities, along with the uncountable family of upper $\alpha$-densities, where $\alpha$ is a real parameter $\ge -1$ (most notably, $\alpha = -1$ corresponds to the upper logarithmic density, and $\alpha = 0$ to the upper asymptotic density). It turns out that a subset of $\bf H$ is small if and only if it belongs to the zero set of the upper Buck density on $\bf Z$. This allows us to show that many interesting sets are small, including the integers with less than a fixed number of prime factors, counted with multiplicity; the numbers represented by a binary quadratic form with integer coefficients whose discriminant is not a perfect square; and the image of $\bf Z$ through a non-linear integral polynomial in one variable.